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Conclusions🪶

✅ To Do

  • Create final tests for conclusions
  • Add results from final tests to conclusions
  • Update Take Home Messages (questions that need answering).

Figure Analysis Overview

  1. Spectral Recovery & Subspace Fidelity: Figures 4 and 7 demonstrate that the model accurately captures the energy eigenvalues (\(E_1 = 1.50\), \(E_2 = 2.40\)), preserves strict spatial orthonormality (\(\langle \hat{\psi}_m^\theta | \hat{\psi}_n^\theta \rangle = \delta_{mn}\)), and accurately spans the low-energy Hilbert subspace (\(\ge 0.96\) alignment with true analytical eigenstates).
  2. Potential Identifiability & Asymptotics: Figure 2 highlights that the inverse problem is under-determined; the optimizer minimizes data loss using an asymmetric sigmoidal potential step rather than the true parabolic well, inducing spurious high-frequency ripples in low-density tails (Figure 3).
  3. Optimization Trajectory: Multi-scale training diagnostics (Figure 1) confirm two distinct transitions (Spike 1 at epoch 5 and spike 2 at epoch 782) followed by an invariant plateau across epochs 800-6000.
  4. POD Basis DecouplingL POD diagnostics (Figures 6 and 8-13) demonstrate that unweighted snapshot SVD produces degenerate singular values (\(\sigma_k = 1.0\)) rotated spatial modes, which can be resolved via physical meausre-weighted decomposition.

Take-Home Messages

Identifiability
  • Q1: Which features of the ground truth potential \(V_\theta(x)\) are uniquely recoverable?
  • Q2: Does smoothness regularization bias the recovered potential family?
Answers
  • A1: The overall shape (curvature) is generally recoverable where probability density \(\rho(x)\) is high. Amplitude and DC offset may be biased by regularization.
  • A2: Yes, it penalizes high-frequency oscillations and tends to produce "flatter" potentials if the loss term weight \(\lambda_\text{smooth}\) is too high, potentially missing sharp features.
Mode Structure
  • Q1: Does orthogonality emerge without reinforcement?
  • Q2: Does POD reveal effective low-rank eigenspaces?
Answers
  • A1: It can weakly emerge through the TISE coupling, but explicit orthonormalization or the energy ordering loss is usually required for stable convergence of multiple states.
  • A2: Yes, the singular value decay indicates the effective dimensionality of the learned wavefunction space.
Inverse Stability
  • Q1: Does noise induce mode mixing?
  • Q2: Are certain eigenstates more stable under low-fidelity observation?
Answers
  • A1: Yes, noise in \(\rho_n^\text{obs}\) can lead to "aliasing" where the model blends the physical eigenstates to fit the noise.
  • A2: Typically, lower-energy states are more stable as they typically have simpler model structure and higher signal-to-noise ratios in many physical systems.
Structural Recovery
  • Q1: Is curvature recoverable before amplitude?
  • Q2: Are nodal locations more identifiable than potential amplitude?
Answers
  • A1: In this architecture, curvature (the second derivative) is an 'operator' that is a major component of the kinetic term of the TISE. Thus, the optimizer is highly sensitive to curvature since both the Schrödinger residual and the smoothness regularizer loss terms depend on second derivatives.
  • A2: Based on the results in the figure analysis, nodal points corresponded with regions of the trained curves that were more aligned with the ground truth wavefunctions. This agrees with the intuition that nodes are more robust to potential fluctuations.

Reproducibility Sanity Check

🏘️ Replicating results with a separate training loop helps reveal any bugs that are masked by expected results.

Normalization Hygiene

🧿️ Do NOT carelessly introduce normalization steps without keeping track. If you make the decision to double-normalize, have a justification and make a note of it.

Rule of Thumb

👍 Normalize the Domain, Constrain the Range

Project 2 Normalization Steps

Step Method Type Justification
1. Grid Setup make_grid Domain Prevents "stiff" gradients originating from extreme input scales
2. Orthonormalization model.psi_theta Range Gram-Schmidt + re-normalization; prevents mode collapse and ensures physical density.
3. Stability \(\epsilon=1e-8\) Range Prevents NaN during initialization
4. Sign Correction Overlap check Alignment Resolves \(\pm \psi_n^\theta(x)\) phase ambiguity for plotting
5. POD Scaling \(1/\sqrt{dx}\) Alignment Maps abstract SVD vectors to physical L2 norms
🔮 Future Implementations
  • Constant \(\Delta x\) implementations:
    • Dirichlet boundary conditions.
    • Simpson's rule quadrature formula in place of the trapezoidal rule.
  • Adaptive collocation.
  • Symplectic loss term and other Hamiltonian-preserving regularizers.
  • Learnable \(\lambda\)'s.
  • Fourier features should increase the MLP's ability to capture high-frequency components.